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We determine non-perturbatively the renormalization constant of the flavour singlet local vector current with O($a$)-improved Wilson fermions. The renormalization constant is fixed by comparing the expectation values (one-point function) of the local vector current and of the conserved one in thermal QCD in a moving reference frame with a non-zero imaginary chemical potential and in the chiral limit. We implement the method in QCD with 3 flavours discretized by the standard Wilson action for gluons and the non-perturbatively O($a$)-improved Wilson fermions. By carrying out extensive numerical simulations, the renormalization constant is determined with a permille precision for values of the bare coupling constant in the range $0.52 \leq g_0^2 \leq 1.13$.

We study the renormalization group (RG) running of the quark mass, for Nf=3 QCD with Wilson fermions in a mixed action setup, with standard Schrödinger functional (SF) boundary conditions for sea quarks and chirally rotated Schrödinger functional (χSF) boundary conditions for valence quarks. This necessitates the tuning of the boundary factor zf(g02) of the χSF valence action, in order to ensure that QCD symmetries are fully recovered in the continuum. The properties of this novel setup are monitored through the ratios ZS/ZP and ςS/ςP of the renormalization parameters and step scaling functions of the scalar and pseudoscalar densities. Where comparison is possible, our ZS/ZP results are found to agree with previous determinations, based on a mass ratio method [G. M. de Divitiis (ALPHA Collaboration), Eur. Phys. J. C 79, 797 (2019)EPCFFB1434-604410.1140/epjc/s10052-019-7287-1] and Ward identities [J. Heitger (ALPHA Collaboration), Eur. Phys. J. C 80, 765 (2020)EPCFFB1434-604410.1140/epjc/s10052-020-8266-2; J. Heitger, Eur. Phys. J. C 81, 606 (2021)EPCFFB1434-604410.1140/epjc/s10052-021-09387-z], with Schrödinger functional boundary conditions. The behavior of ςS/ςP confirms the theoretical expectations of χSF QCD, related to the restoration of the theory's symmetries in the continuum limit. From the step-scaling function of the pseudoscalar density we obtain the quark mass RG-running function from hadronic to perturbative energy scales. This is fully compatible with the earlier result obtained in a similar setup for Wilson quarks with Schrödinger functional boundary conditions [I. Campos, Eur. Phys. J. C 78, 387 (2018)EPCFFB1434-604410.1140/epjc/s10052-018-5870-5], and provides a strong universality test for the two lattice setups.

The hadronic contribution to the muon anomalous magnetic moment $a_��=(g_��-2)/2$ has to be determined at the per-mille level for the Standard Model prediction to match the expected final uncertainty of the ongoing E989 experiment. That is 3 times better than the current precision from the dispersive approach, and 5-15 times smaller than the uncertainty based on the purely theoretical determinations from lattice QCD. So far the stumbling-block is the large statistical error in the Monte Carlo evaluation of the required correlation functions which can hardly be tamed by brute force. In this talk we present our proposal to solve this problem by multi-level Monte Carlo integration, a technique which reduces the variance of correlators exponentially in the distance of the fields. We report the results of our feasibility tests for the computation of the Hadronic Vacuum Polarization on a lattice with a linear extension of 3~fm, a spacing of 0.065 fm, and a pion mass of 270 MeV. Indeed the two-level integration makes the contribution to the statistical error from long-distances de-facto negligible by accelerating its inverse scaling with the cost of the simulation. These findings establish multi-level Monte Carlo as a solid and efficient method for a precise lattice determination of the hadronic contribution to $a_��$., Proceedings of the 40th International Conference on High Energy physics (ICHEP2020). arXiv admin note: substantial text overlap with arXiv:2007.02973

A summary of the main points of discussion raised during the talks and their follow-up questions, as well as in the round table of the last day of the αs(2019) workshop, is presented. The discussions not only focused on particular issues affecting each one of the individual αs extractions, but also on the current PDG categorization of αs measurements and on the methods used to average them into a single αs(mZ) value. Most of the listed points are open and sources of potential controversies, which we highlight here as one might expect that ongoing progress in the field will lead to their clarification and resolution.

Using continuum extrapolated lattice data we trace a family of running couplings in three-flavour QCD over a large range of scales from about 4 to 128 GeV. The scale is set by the finite space time volume so that recursive finite size techniques can be applied, and Schrödinger functional (SF) boundary conditions enable direct simulations in the chiral limit. Compared to earlier studies we have improved on both statistical and systematic errors. Using the SF coupling to implicitly define a reference scale 1 / L0≈ 4 GeV through g¯ 2(L0) = 2.012 , we quote L0ΛMS¯Nf=3=0.0791(21). This error is dominated by statistics; in particular, the remnant perturbative uncertainty is negligible and very well controlled, by connecting to infinite renormalization scale from different scales 2 n/ L0 for n= 0 , 1 , … , 5. An intermediate step in this connection may involve any member of a one-parameter family of SF couplings. This provides an excellent opportunity for tests of perturbation theory some of which have been published in a letter (ALPHA collaboration, M. Dalla Brida et al. in Phys Rev Lett 117(18):182001, 2016). The results indicate that for our target precision of 3 per cent in L0ΛMS¯Nf=3, a reliable estimate of the truncation error requires non-perturbative data for a sufficiently large range of values of αs= g¯ 2/ (4 π). In the present work we reach this precision by studying scales that vary by a factor 2 5= 32 , reaching down to αs≈ 0.1. We here provide the details of our analysis and an extended discussion.

We investigate the quark content of the low-lying states in the I(JP)=1(0+) sector, which are the quantum numbers of the a0(980) meson, using lattice QCD. To this end, we consider correlation functions of six different two- and four-quark interpolating fields. We evaluate all diagrams, including diagrams, where quarks propagate within a time slice, e.g. with closed quark loops. We demonstrate that diagrams containing such closed quark loops have a drastic effect on the final results and, thus, may not be neglected. Our analysis, which is carried out at unphysically heavy u and d quark mass corresponding to mπ=296(3) MeV and in a single spatial volume of extent 2.9 fm, shows that in addition to the expected spectrum of two-meson scattering states there is an additional energy level around the two-particle thresholds of K+K̄ and η+π. This additional state, which is a candidate for the a0(980) meson, couples to a quark-antiquark as well as to a diquark-antidiquark interpolating field, indicating that it is a superposition of an ordinary q̄q and a tetraquark structure. The analysis is performed using AMIAS, a novel statistical method based on the sampling of all possible spectral decompositions of the considered correlation functions, as well as solving standard generalized eigenvalue problems.

We present results by the ALPHA collaboration for the L-parameter in 3-flavour QCD and the strong coupling constant at the electroweak scale, αs(mZ), in terms of hadronic quantities computed on the CLS gauge configurations. The first part of this proceedings contribution contains a review of published material [1, 2] and yields the L-parameter in units of a low energy scale, 1=Lhad. We then discuss how to determine this scale in physical units from experimental data for the pion and kaon decay constants. We obtain Λ(3)/MS = 332(14)MeV which translates to αs(MZ) = 0:1179(10)(2) if one uses the standard perturbative procedure to match between 3-, 4- and 5-flavour QCD.

Using a finite volume gradient flow renormalization scheme with Schrödinger Functional boundary conditions, we compute the nonperturbative running coupling in the range 2.2gGF2(L)13. Careful continuum extrapolations turn out to be crucial to reach our high accuracy. The running of the coupling is always between one loop and two loop and very close to one loop in the region of 200 MeVμ=1/L4 GeV. While there is no convincing contact to two-loop running, we match nonperturbatively to the Schrödinger functional coupling with background field. In this case, we know the μ-dependence up to ∼100 GeV and can thus connect to the Λ-parameter

In this contribution we present an exploratory study of several novel methods for numerical stochastic perturbation theory. For the investigation we consider observables defined through the gradient flow in the simple φ4 theory.

To investigate the light scalar tetraquark candidate a0(980) (quantum numbers JP =0+), a correlation matrix including a variety of two- and four-quark interpolating operators has to be computed. We discuss efficient techniques to compute the elements of this correlation matrix, in particular diagrams with closed quark loops. Furthermore, we present evidence that such diagrams are not negligible given our precision, and their contribution is essential to obtain physically meaningful results. In particular, we find indications of the existence of an "additional" state around the two-particle thresholds of K+ K̄ and η +π, which could correspond to the a0(980) meson. © Copyright owned by the author(s). Part F128557 Sponsors: Conference code: 128557

We explore a method to extract energy eigenstates, called Athens Model Independent Analysis Scheme (AMIAS), which is an alternative to solving standard Generalized Eigenvalue Problems (GEVP). The method is based on statistically sampling the space of fit parameters according to the χ2 value of the fit function. The method is particularly suited for correlators or correlation matrices with strong contributions from several energy eigenstates and for rather noisy data, e.g. for correlators with disconnected and partly disconnected diagrams. We apply the method to the analysis of the JP =0+ channel in the context of our investigation of the a0(980) meson and point out advantages compared to the GEVP. © Copyright owned by the author(s). Part F128557 Sponsors: Conference code: 128557

We review the status of an ongoing long-term lattice investigation of the spectrum and structure of tetraquark candidates. We focus on the light scalar meson $a_0(980)$. First steps regarding the study of a possibly existing $c c \bar{c} \bar{c}$ tetraquark are also outlined., 5 pages, 3 figures, talk given at FAIRNESS 2013, September 15-21 2013, Berlin, Germany

We carry out an exploratory study of the isospin one a0(980) and the isospin one-half kappa scalar mesons using Nf=2+1+1 Wilson twisted mass fermions at one lattice spacing. The valence strange quark is included as an Osterwalder-Seiler fermion with mass tuned so that the kaon mass matches the corresponding mass in the unitary Nf=2+1+1 theory. We investigate the internal structure of these mesons by using a basis of four-quark interpolating fields. We construct diquark-diquark and molecular-typecinterpolating fields and analyse the resulting correlation matrices keeping only connected contributions. For both channels, the low-lying spectrum is found to be consistent with two-particle scattering states. Therefore, our analysis shows no evidence for an additional state that can be interpreted as either a tetraquark or a tightly-bound molecular state., Comment: 5 figures, 23 pages

The conjecture that several recently observed mesons have a structure, which is not dominated by an ordinary quark-Antiquark pair, but by a four-quark structure, is being actively investigated both theoretical and experimentally. Such a state may be characterized as a mesonic molecule or as a diquark-Antidiquark pair. Lattice QCD provides a theoretically sound framework to study such states. To quantitatively investigate the internal structure of such mesons, one needs to precisely compute correlation matrices containing several interpolating operators including two and four quarks. Here we discuss certain technical aspects of such correlation matrices suited to study tetraquark candidates with Jp = 0+ and flavor structure q1q2q3q3, e.g. the a0(980) meson, the D∗S0 meson and some of the charged cc X states. Some numerical results for the a0(980) meson are presented. 14-18-July-2015 Sponsors: Conference code: 123491